In the end I decided to reduce the existing double socket to a single and it seems to be okay:
I think there's just about enough space between the single socket and the new radiator to not cause an issue.
If I did decide later, I could install a new double socket behind the bedside cabinet and run a couple of new cables from this horizontally to the single socket to extend the ring main, replacing the single socket with a blank face plate to provide access to the connections as per the regulations.
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